Faulty picture-hanging improved

Abstract

A picture-hanging puzzle is the task of hanging a framed picture with a wire around a set of nails in such a way that it can remain hanging on certain specified sets of nails, but will fall if any more are removed. The classical brain teaser asks us to hang a picture on two nails in such a way that it falls when any one is detached. Demaine et al (2012) proved that all reasonable puzzles of this kind are solvable, and that for the $k$-out-of-$n$ problem, the size of a solution can be bounded by a polynomial in $n$. We give simplified proofs of these facts, for the latter leading to a reasonable exponent in the polynomial bound.